Bayesian networks are a tool for modeling systems. A description of Bayesian networks is provided in U.S. Pat. No. 6,408,290, which description is provided below, with omissions indicated by ellipses. FIG. 1 from the U.S. Pat. No. 6,408,290 patent is replicated as FIG. 1 hereto:                A Bayesian network is a representation of the probabilistic relationships among distinctions about the world. Each distinction, sometimes called a variable, can take on one of a mutually exclusive and exhaustive set of possible states. A Baybsian network is expressed as an acyclic-directed graph where the variables correspond to nodes and the relationships between the nodes correspond to arcs. FIG. 1 depicts an exemplary Bayesian network 101. In FIG. 1 there are three variables, X1, X2, and X3, which are represented by nodes 102, 106 and 110, respectively. This Bayesian network contains two arcs 104 and 108. Associated with each variable in a Bayesian network is a set of probability distributions. Using conditional probability notation, the set of probability distributions for a variable can be denoted by p(xi|Πi, ζ) where “p” refers to the probability distribution, where “Πi” denotes the parents of variable Xi and where “ζ” denotes the knowledge of the expert. The Greek letter “ζ” indicates that the Bayesian network reflects the knowledge of an expert in a given field. Thus, this expression reads as follows: the probability distribution for variable Xi given the parents of Xi and the knowledge of the expert. For example, X1 is the parent of X2. The probability distributions specify the strength of the relationships between variables. For instance, if X1 has two states (true and false), then associated with X1 is a single probability distribution p(xi|ζ) and associated with X2 are two probability distributions p(xi|x1=t,ζ) and p(xi|x1=f ,ζ). . . .        The arcs in a Bayesian network convey dependence between nodes. When there is an arc between two nodes, the probability distribution of the first node depends upon the value of the second node when the direction of the arc points from the second node to the first node. For example, node 106 depends upon node 102. Therefore, nodes 102 and 106 are said to be conditionally dependent. Missing arcs in a Bayesian network convey conditional independencies. For example, node 102 and node 110 are conditionally independent given node 106. However, two variables indirectly connected through intermediate variables are conditionally dependent given lack of knowledge of the values (“states”) of the intermediate variables. Therefore, if the value for node 106 is known, node 102 and node 110 are conditionally dependent.        In other words, sets of variables X and Y are said to be conditionally independent, given a set of variables Z, if the probability distribution for X given Z does not depend on Y. If Z is empty, however, X and Y are said to be “independent” as opposed to conditionally independent. If X and Y are not conditionally independent, given Z, then X and Y are said to be conditionally dependent given Z.        The variables used for each node may be of different types. Specifically, variables may be of two types: discrete or continuous. A discrete variable is a variable that has a finite or countable number of states, whereas a continuous variable is a variable that has an uncountably infinite number of states. . . . An example of a discrete variable is a Boolean variable. Such a variable can assume only one of two states: “true” or “false.” An example of a continuous variable is a variable that may assume any real value between −1 and 1. Discrete variables have an associated probability distribution. Continuous variables, however, have an associated probability density function (“density”). Where an event is a set of possible outcomes, the density p(x) for a variable “x” and events “a” and “b” is defined as:        
      p    ⁡          (      x      )        =            Lim              a        →        b              ⁡          [                        p          ⁡                      (                          a              ≤              x              ≤              b                        )                                                          (                          a              -              b                        )                                        ]                      where p(a≦x≦b) is the probability that x lies between a and b.        
Bayesian networks also make use of Bayes Rule, which states:
      p    ⁡          (              B        |        A            )        =                    p        ⁡                  (          B          )                    ·              p        ⁡                  (                      A            |            B                    )                            p      ⁡              (        A        )            for two variables, where p(B|A) is sometimes called an a posteriori probability. Similar equations have been derived for more than two variables. The set of all variables associated with a system is known as the domain.
Building a network with the nodes related by Bayes Rule allows changes in the value of variables associated with a particular node to ripple through the probabilities in the network. For example, referring to FIG. 1, assuming that X1, X2 and X3 have probability distributions and that each of the probability distributions is related by Bayes Rule to those to which it is connected by arcs, then a change to the probability distribution of X2 may cause a change in the probability distribution of X1 (through induction) and X3 (through deduction). Those mechanisms also establish a full joint probability of all domain variables (i.e. X1, X2, X3) while allowing the data associated with each variable to be uncertain.
Software available from several vendors, including Norsys Software Corp., allows construction of Bayesian networks with many interrelated nodes, with each node capable of having many states. Bayesian networks may be used to construct Bayesian Belief Networks and Bayesian Decision Networks.
As mentioned above, Bayesian networks have been used to model physical systems. In some cases, models of physical systems are based on observed data. Frequently, such data has a quality, or reliability, parameter associated with it that reflects the amount of confidence an observer has in the observed data.